Despite its long history, magnetism remains at the forefront of fundamental research in condensed matter physics. This dissertation addresses three interrelated projects that explore key phenomena in magnetism, including geometric frustration in fcc antiferromagnets, the magnetocaloric effect in ferromagnetic materials, and metamagnetism in the superconducting heavy-fermion material URhGe. These topics represent distinct but connected areas of study that reveal important insights into the behavior of magnetic systems and their phase transitions.

The first part of this dissertation investigates geometric frustration in face-centered cubic (fcc) antiferromagnets, with a particular focus on the special ratio of exchange constants, which enhances frustration in the system. It leads to an octahedra-block representation of the original spin Hamiltonian, resulting in a strongly frustrated spin model. By employing Monte Carlo simulations, a surprisingly rich magnetic phase diagram was obtained. In particular, this spin model exhibits two magnetization plateaus.

The second project investigates the metamagnetic behavior of the superconducting heavy-fermion material URhGe, a system with unique magnetic properties. URhGe is a ferromagnetic superconductor that exhibits a distinctive magnetic behavior. When a magnetic field is applied along the b direction, URhGe undergoes an abrupt orientational transition of the magnetization, with a reentrant superconducting phase emerging close to the transition field. This behavior is particularly fascinating in the context of the interplay between magnetism and superconductivity. To explore the magnetic properties of URhGe, a theoretical spin model with competing magnetic anisotropies was developed. The model was analyzed both analytically at zero temperature and using Monte Carlo simulations at finite temperatures. The constructed phase diagram features a tricritical point and shows excellent quantitative agreement with the experimental phase diagram of URhGe. The study demonstrates that the asymptotic tricritical behavior of the order parameter and the correlation length is governed by mean-field critical exponents. This part of the dissertation contributes to understanding the metamagnetic transition in heavy-fermion superconductors and the role of magnetic anisotropies in shaping the phase diagrams of such complex materials.

The last project focuses on the magnetocaloric effect (MCE) in ferromagnetic materials, which is a thermodynamic phenomenon where a material's temperature changes in response to an applied magnetic field. The MCE is of significant interest both for fundamental research and practical applications such as magnetic refrigeration. Using Monte Carlo methods, the MCE was studied in ferromagnetic gadolinium.
The results were compared with available experimental data, showing a good agreement between the simulations and observations. This work demonstrates that Monte Carlo simulations can be used not only to study phase transitions and critical phenomena but also to model thermodynamic effects, such as the magnetocaloric effect, with high precision. The findings further establish the versatility of Monte Carlo methods in the study of magnetism and thermodynamic properties of magnetic materials.

The work presented in this dissertation highlights the broad applicability of Monte Carlo simulations for studying magnetic materials, from conventional ferromagnets to more exotic systems like frustrated antiferromagnets. The results presented here contribute to a deeper understanding of the physics of magnetism and open up new possibilities for future research in condensed matter physics.​